| L(s) = 1 | + (12.2 + 10.2i)2-s + (−14.6 + 5.32i)3-s + (44.4 + 252. i)4-s + (370. − 2.09e3i)5-s + (−234. − 85.2i)6-s + (−1.06e3 − 1.84e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (−1.48e4 + 1.24e4i)9-s + (2.61e4 − 2.19e4i)10-s + (−4.51e4 + 7.81e4i)11-s + (−1.99e3 − 3.45e3i)12-s + (−5.20e4 − 1.89e4i)13-s + (5.91e3 − 3.35e4i)14-s + (5.76e3 + 3.26e4i)15-s + (−6.15e4 + 2.24e4i)16-s + (−2.72e5 − 2.28e5i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.104 + 0.0379i)3-s + (0.0868 + 0.492i)4-s + (0.264 − 1.50i)5-s + (−0.0737 − 0.0268i)6-s + (−0.167 − 0.290i)7-s + (−0.176 + 0.306i)8-s + (−0.756 + 0.634i)9-s + (0.826 − 0.693i)10-s + (−0.929 + 1.60i)11-s + (−0.0277 − 0.0480i)12-s + (−0.505 − 0.183i)13-s + (0.0411 − 0.233i)14-s + (0.0294 + 0.166i)15-s + (−0.234 + 0.0855i)16-s + (−0.792 − 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0137276 - 0.0809769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0137276 - 0.0809769i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-12.2 - 10.2i)T \) |
| 19 | \( 1 + (2.69e5 + 4.99e5i)T \) |
| good | 3 | \( 1 + (14.6 - 5.32i)T + (1.50e4 - 1.26e4i)T^{2} \) |
| 5 | \( 1 + (-370. + 2.09e3i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (1.06e3 + 1.84e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.51e4 - 7.81e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (5.20e4 + 1.89e4i)T + (8.12e9 + 6.81e9i)T^{2} \) |
| 17 | \( 1 + (2.72e5 + 2.28e5i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-2.23e5 - 1.26e6i)T + (-1.69e12 + 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-1.03e6 + 8.64e5i)T + (2.51e12 - 1.42e13i)T^{2} \) |
| 31 | \( 1 + (1.56e6 + 2.71e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + 5.04e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.41e7 - 8.77e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (4.52e6 - 2.56e7i)T + (-4.72e14 - 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-1.86e7 + 1.56e7i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (1.71e7 + 9.70e7i)T + (-3.10e15 + 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-5.94e7 - 4.98e7i)T + (1.50e15 + 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-2.63e7 - 1.49e8i)T + (-1.09e16 + 3.99e15i)T^{2} \) |
| 67 | \( 1 + (-1.62e8 + 1.35e8i)T + (4.72e15 - 2.67e16i)T^{2} \) |
| 71 | \( 1 + (2.97e7 - 1.68e8i)T + (-4.30e16 - 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-3.16e7 + 1.15e7i)T + (4.50e16 - 3.78e16i)T^{2} \) |
| 79 | \( 1 + (4.99e8 - 1.81e8i)T + (9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (3.26e8 + 5.65e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-7.94e8 - 2.89e8i)T + (2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (3.54e8 + 2.97e8i)T + (1.32e17 + 7.48e17i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.42917913745168856836244440725, −12.94721055432473555043503626210, −11.65350453925044816546278068488, −9.865558585216885606101174423020, −8.524282396826272605812597891471, −7.21306187792430103450541612561, −5.25530812886009786733003049662, −4.66806351890244703808505422231, −2.25454975053571689191975961045, −0.02272209585294683125908863636,
2.47580624905117260678607612671, 3.43093217318587087564244079465, 5.71769919308421159908515334487, 6.61689400817894233883779862880, 8.614326444717813639606155149090, 10.43792076930652922141622294844, 11.03797193716032980183589628893, 12.37125456400055551990164502769, 13.82785068196085792675752452735, 14.55885166410014693742361564504
